Uniformly Continuous

Let and be Metric Spaces.
A function is uniformly continuous if

Lemma

If is uniformly continuous then it is Continuous.

Theorem

If is continuous on a closed interval
then it is uniformly continuous on .

Proof sketch

Suppose otherwise.
Choose a “bad” and , along with .
Use BWT to find convergent (which has to stay in the closed interval).
Then also .
Now find a contradiction with continuity at .

Theorem

If is continuous on then it is Riemann Integrable on .

Proof sketch

By previous theorem, is uniformly continuous.
Now choose a dissection s.t. .
Write out and and bind them appropriately - done.